Testing strong gravitational lensing effects of various supermassive compact objects for the static and spherically symmetric hairy black hole by gravitational decoupling

Abstract We investigate the strong gravitational lensing effects of various supermassive compact objects for the static and spherically symmetric hairy black hole by gravitational decoupling. To do this purpose, we first numerically calculate the strong deflection limit coefficients and strong deflection angle with the decoupling modified black hole parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> and $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> . It is observed that for the lowest value of the parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> , the strong deflection angle $$\alpha _D$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>α</mml:mi><mml:mi>D</mml:mi></mml:msub></mml:math> reaches its maximum value with the parameter $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> and for the lowest value of the parameter $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> , strong deflection angle $$\alpha _D$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>α</mml:mi><mml:mi>D</mml:mi></mml:msub></mml:math> reaches its minimum value with the parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> ; and also the strong deflection angle for $$\alpha _D$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>α</mml:mi><mml:mi>D</mml:mi></mml:msub></mml:math> for the decoupling modified black hole is smaller than standard Schwarzschild black hole $$(\alpha =0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> . We investigate the various astrophysical consequences via strong gravitational lensing by considering the example of various supermassive BHs in the centre nearby galaxies and observe that the decoupling modified black hole could be quantitatively distinguished from the standard Schwarzschild black hole ( $$\alpha =0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> ). Findings in our observation suggest that the observational test for a decoupling-modified black hole is indeed feasible.

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